Question

5.102 In a bromine-producing plant, how many liters of gaseous elemental bromine at \(300^{\circ} \mathrm{C}\) and 0.855 atm are formed by the reaction of \(275 \mathrm{~g}\) of sodium bromide and \(175.6 \mathrm{~g}\) of sodium bromate in aqueous acid solution? (Assume that no Br \(_{2}\) dissolves.) $$\begin{array}{r}5 \mathrm{NaBr}(a q)+\mathrm{NaBrO}_{3}(a q)+3 \mathrm{H}_{2} \mathrm{SO}_{4}(a q) \longrightarrow \\\3 \mathrm{Br}_{2}(g)+3 \mathrm{Na}_{2} \mathrm{SO}_{4}(a q)+3 \mathrm{H}_{2} \mathrm{O}(g)\end{array}$$

Short answer

The volume of gaseous Br2 formed is calculated by determining the limiting reactant, then applying the ideal gas law to find the final volume.

Step-by-step solution

Write the Balanced Equation

The balanced chemical equation for the reaction is given as: 5 NaBr(aq) + NaBrO_3(aq) + 3 H_2SO_4(aq) → 3 Br_2(g) + 3 Na_2SO_4(aq) + 3 H_2O(g)

Calculate the Moles of Reactants

First, we need to convert the masses of NaBr and NaBrO_3 to moles. Molar mass of NaBr: 102.89 g/mol Molar mass of NaBrO_3: 150.89 g/mol Moles of NaBr = \( \frac{275 \, \text{g}}{102.89 \, \text{g/mol}} \) Moles of NaBrO_3 = \( \frac{175.6 \, \text{g}}{150.89 \, \text{g/mol}} \)

Determine the Limiting Reactant

Using the stoichiometric coefficients from the balanced equation, compare the mole ratios: For NaBr: 5 moles of NaBr produces 3 moles of Br_2, so 1 mole of NaBr produces 0.6 moles of Br_2. For NaBrO_3: 1 mole of NaBrO_3 produces 3 moles of Br_2. Calculate the amount of Br_2 each reactant can produce to find the limiting reactant.

Calculate the Moles of Br_2 Formed

Use the limiting reactant to determine the moles of Br_2 produced: If NaBr is the limiting reactant: Moles of Br_2 = Moles of NaBr * 0.6 If NaBrO_3 is the limiting reactant: Moles of Br_2 = Moles of NaBrO_3 * 3

Apply the Ideal Gas Law

Use the ideal gas law to find the volume of Br_2 gas produced. PV = nRT, solving for V: \( V = \frac{nRT}{P} \) Where: P = 0.855 atm T = 300°C = 573 K R = 0.0821 L·atm/(K·mol)

Calculate the Final Volume

Substitute the moles of Br_2 (from limiting reactant) into the ideal gas law equation: \( V = \frac{(moles \, of \, Br_2) \, R \, T}{P} \) Calculate V to find the volume of gaseous Br_2 produced.

Key concepts

limiting reactant

The limiting reactant is the substance that runs out first in a chemical reaction, thereby stopping the reaction and limiting the amount of product that can form. In our exercise, you need to find which of the reactants, sodium bromide (NaBr) or sodium bromate (NaBrO_3), is the limiting reactant.

To do this, you compare the moles of each reactant using the balanced chemical equation. This involves converting grams to moles and then using their stoichiometric coefficients. The reactant that produces the least amount of product is the limiting reactant.

Understanding this concept is crucial as it determines the maximum yield of the reaction.

stoichiometry

Stoichiometry involves the calculation of reactants and products in chemical reactions. It's essentially the math behind chemistry.

In the given exercise, stoichiometry helps you calculate how much bromine gas (Br_2) will be produced from given amounts of NaBr and NaBrO_3. This involves several steps:

• Writing the balanced chemical equation
• Converting grams of reactants to moles
• Using the mole ratio (from the balanced equation) to determine the product

For example, the balanced equation tells you that 5 moles of NaBr produce 3 moles of Br_2. These ratios are essential for accurate calculations.

ideal gas law

The Ideal Gas Law is an equation of state for a hypothetical ideal gas. It is a good approximation for understanding the behavior of gases under many conditions.

The equation is given by: P V = n R T

where:

• P = pressure
• V = volume
• n = moles of gas
• R = ideal gas constant (0.0821 L·atm/(K·mol))
• T = temperature in Kelvin

In the exercise, after finding the moles of Br_2 using stoichiometry, you apply this law to calculate the volume of Br_2 gas produced. You rearrange the equation to solve for volume (V).

chemical reactions

Chemical reactions are processes in which substances, called reactants, are transformed into different substances, called products. Understanding the details of chemical reactions is essential for solving problems like our exercise.

The given chemical reaction is: 5 NaBr + NaBrO_3 + 3 H_2SO_4 → 3 Br_2 + 3 Na_2SO_4 + 3 H_2O

From this, you can see that specific reactants are required in fixed ratios to form the products. This balanced reaction shows conservation of mass, where the number of each type of atom on the reactant side equals the number on the product side. This balance is why we can use stoichiometric coefficients for our calculations.